# Multiresolution local smoothness detection in non-uniformly sampled multivariate signals

arXiv: https://arxiv.org/abs/2507.13480 · HTML: https://mlsanity.com/paper/2507.13480/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 33 references that are in the map.
- First posted on arXiv 17 Jul 2025.

## Builds on (most cited first, 10 of 10)

- [Holistically-Nested Edge Detection](https://mlsanity.com/paper/1504.06375/) (2015)
- [DeepEdge: A Multi-Scale Bifurcated Deep Network for Top-Down Contour Detection](https://mlsanity.com/paper/1412.1123/) (2014)
- [Dense Extreme Inception Network for Edge Detection](https://mlsanity.com/paper/2112.02250/) (2021)
- [ N4 -Fields: Neural Network Nearest Neighbor Fields for Image Transforms](https://mlsanity.com/paper/1406.6558/) (2014)
- [Pixel-wise Deep Learning for Contour Detection](https://mlsanity.com/paper/1504.01989/) (2015)
- [Multiresolution kernel matrix algebra](https://mlsanity.com/paper/2211.11681/) (2022)
- [Anisotropic multiresolution analyses for deepfake detection](https://mlsanity.com/paper/2210.14874/) (2022)
- [Samplet basis pursuit: Multiresolution scattered data approximation with sparsity constraints](https://mlsanity.com/paper/2306.10180/) (2023)
- [Multiscale scattered data analysis in samplet coordinates](https://mlsanity.com/paper/2409.14791/) (2024)
- [Construction of generalized samplets in Banach spaces](https://mlsanity.com/paper/2412.00954/) (2024)

## Built on by (most cited first, 2 of 2)

- [Bespoke multiresolution analysis of graph signals](https://mlsanity.com/paper/2507.19181/) (2025)
- [Data-intrinsic approximation in metric spaces](https://mlsanity.com/paper/2510.13496/) (2025)

## Abstract (arXiv)

Inspired by edge detection based on the decay behavior of wavelet coefficients, we introduce a (near) linear-time algorithm for detecting the local regularity in non-uniformly sampled multivariate signals. Our approach quantifies regularity within the framework of microlocal spaces introduced by Jaffard. The central tool in our analysis is the fast samplet transform, a distributional wavelet transform tailored to scattered data. We establish a connection between the decay of samplet coefficients and the pointwise regularity of multivariate signals. As a by product, we derive decay estimates for functions belonging to classical H"older spaces and Sobolev-Slobodeckij spaces. While traditional wavelets are effective for regularity detection in low-dimensional structured data, samplets demonstrate robust performance even for higher dimensional and scattered data. To illustrate our theoretical findings, we present extensive numerical studies detecting local regularity of one-, two- and three-dimensional signals, ranging from non-uniformly sampled time series over image segmentation to edge detection in point clouds.